On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

The Bipartite-Cylindrical Crossing Number of the Complete Bipartite Graph

2019 ◽  
Vol 36 (2) ◽  
pp. 205-220
Author(s):  
Bernardo Ábrego ◽  
Silvia Fernández-Merchant ◽  
Athena Sparks
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

scholarly journals A lower bound for the Graver complexity of the incidence matrix of a complete bipartite graph

2012 ◽  
Vol 3 (4) ◽  
pp. 695-708 ◽  
Author(s):  
Taisei Kudo ◽  
Akimichi Takemura
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Incidence Matrix ◽  
Complete Bipartite Graph ◽  
Graver Complexity

TENACITY OF TOTAL GRAPHS

2014 ◽  
Vol 25 (05) ◽  
pp. 553-562 ◽  
Author(s):  
YINKUI LI ◽  
ZONGTIAN WEI ◽  
XIAOKUI YUE ◽  
ERQIANG LIU
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Communication Networks ◽  
Interconnection Networks ◽  
Complete Bipartite Graph ◽  
Total Graph ◽  
The Stability

Communication networks must be constructed to be as stable as possible, not only with the respect to the initial disruption, but also with respect to the possible reconstruction. Many graph theoretical parameters have been used to describe the stability of communication networks. Tenacity is a reasonable one, which shows not only the difficulty to break down the network but also the damage that has been caused. Total graphs are the largest graphs formed by the adjacent relations of elements of a graph. Thus, total graphs are highly recommended for the design of interconnection networks. In this paper, we determine the tenacity of the total graph of a path, cycle and complete bipartite graph, and thus give a lower bound of the tenacity for the total graph of a graph.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

Generalization of the Symmetric Starter of the Orthogonal Double Covers of the Complete Bipartite Graph

2006 ◽  
Vol 16 (2) ◽  
pp. 171-177
Author(s):  
S. A. El-Serafi ◽  
R. A. El-Shanawany ◽  
M. Sh. Higazy
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Double Covers

scholarly journals An Improvement of the Lower Bound on the Minimum Number of ≤k-Edges

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 525
Author(s):  
Javier Rodrigo ◽  
Susana Merchán ◽  
Danilo Magistrali ◽  
Mariló López
Keyword(s):  
Lower Bound ◽  
Complete Graph ◽  
General Position ◽  
Crossing Number ◽  
Minimum Number

In this paper, we improve the lower bound on the minimum number of  ≤k-edges in sets of n points in general position in the plane when k is close to n2. As a consequence, we improve the current best lower bound of the rectilinear crossing number of the complete graph Kn for some values of n.

Spanning Trees of the Complete Bipartite Graph

1990 ◽  
pp. 339-346 ◽  
Author(s):  
N. Hartsfield ◽  
J. S. Werth
Keyword(s):  
Bipartite Graph ◽  
Spanning Trees ◽  
Complete Bipartite Graph
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